C2xD5:M4(2) - GroupNames

G = C₂×D₅⋊M₄(2) order 320 = 2⁶·5

Direct product of C₂ and D₅⋊M₄(2)

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂×D₅⋊M₄(2), D₁₀⋊₁₁M₄(2), Dic₅.₁₆C₂⁴, C₅⋊C₈⋊₂C₂³, D₅⋊(C₂×M₄(2)), D₅⋊C₈⋊₉C₂², C₂.₅(C₂³×F₅), C₁₀⋊₂(C₂×M₄(2)), C₁₀.₃(C₂³×C₄), C₄.F₅⋊₁₃C₂², C₅⋊₂(C₂²×M₄(2)), (C₂²×C₄).₂₅F₅, C₂³.₅₃(C₂×F₅), C₄.₅₈(C₂²×F₅), (C₂²×C₂₀).₂₉C₄, C₂₀.₉₈(C₂²×C₄), (C₂³×D₅).₁₉C₄, (C₄×D₅).₉₁C₂³, C₂².F₅⋊₇C₂², D₁₀.₄₅(C₂²×C₄), C₂².₁₉(C₂²×F₅), Dic₅.₄₅(C₂²×C₄), (C₂×Dic₅).₃₆₃C₂³, (C₂²×Dic₅).₂₈₃C₂², (C₂×C₄×D₅).₄₀C₄, (C₂×D₅⋊C₈)⋊₁₃C₂, (C₂×C₅⋊C₈)⋊₁₀C₂², (C₂×C₄.F₅)⋊₁₄C₂, (C₄×D₅).₉₇(C₂×C₄), (C₂×C₄).₁₇₃(C₂×F₅), (D₅×C₂²×C₄).₃₂C₂, (C₂×C₂₀).₁₅₂(C₂×C₄), (C₂×C₄×D₅).₄₀₅C₂², (C₂×C₂².F₅)⋊₁₂C₂, (C₂²×C₁₀).₇₉(C₂×C₄), (C₂×C₁₀).₉₇(C₂²×C₄), (C₂×Dic₅).₁₉₉(C₂×C₄), (C₂²×D₅).₁₃₃(C₂×C₄), SmallGroup(320,1589)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₀ — C₂×D₅⋊M₄(2)

Chief series

C₁ — C₅ — C₁₀ — Dic₅ — C₅⋊C₈ — C₂×C₅⋊C₈ — C₂×D₅⋊C₈ — C₂×D₅⋊M₄(2)

Lower central

C₅ — C₁₀ — C₂×D₅⋊M₄(2)

Upper central

C₁ — C₂×C₄ — C₂²×C₄

Generators and relations for C₂×D₅⋊M₄(2)
G = < a,b,c,d,e | a²=b⁵=c²=d⁸=e²=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, dbd^-1=b³, be=eb, dcd^-1=b²c, ce=ec, ede=d⁵ >

Subgroups: 906 in 298 conjugacy classes, 148 normal (28 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, C₂×C₄, C₂³, C₂³, D₅, D₅, C₁₀, C₁₀, C₁₀, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂⁴, Dic₅, Dic₅, C₂₀, D₁₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, C₂²×C₈, C₂×M₄(2), C₂³×C₄, C₅⋊C₈, C₄×D₅, C₂×Dic₅, C₂×Dic₅, C₂×C₂₀, C₂×C₂₀, C₂²×D₅, C₂²×D₅, C₂²×D₅, C₂²×C₁₀, C₂²×M₄(2), D₅⋊C₈, C₄.F₅, C₂×C₅⋊C₈, C₂².F₅, C₂×C₄×D₅, C₂×C₄×D₅, C₂²×Dic₅, C₂²×C₂₀, C₂³×D₅, C₂×D₅⋊C₈, C₂×C₄.F₅, D₅⋊M₄(2), C₂×C₂².F₅, D₅×C₂²×C₄, C₂×D₅⋊M₄(2)
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, C₂³, M₄(2), C₂²×C₄, C₂⁴, F₅, C₂×M₄(2), C₂³×C₄, C₂×F₅, C₂²×M₄(2), C₂²×F₅, D₅⋊M₄(2), C₂³×F₅, C₂×D₅⋊M₄(2)

Smallest permutation representation of C₂×D₅⋊M₄(2)
►On 80 points

Generators in S₈₀

(1 79)(2 80)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)(25 70)(26 71)(27 72)(28 65)(29 66)(30 67)(31 68)(32 69)(33 47)(34 48)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)(49 59)(50 60)(51 61)(52 62)(53 63)(54 64)(55 57)(56 58)

(1 58 15 40 67)(2 33 59 68 16)(3 69 34 9 60)(4 10 70 61 35)(5 62 11 36 71)(6 37 63 72 12)(7 65 38 13 64)(8 14 66 57 39)(17 29 55 45 78)(18 46 30 79 56)(19 80 47 49 31)(20 50 73 32 48)(21 25 51 41 74)(22 42 26 75 52)(23 76 43 53 27)(24 54 77 28 44)

(1 71)(2 12)(3 64)(4 39)(5 67)(6 16)(7 60)(8 35)(9 65)(10 57)(11 15)(13 69)(14 61)(17 51)(18 22)(19 76)(20 28)(21 55)(23 80)(24 32)(25 29)(26 79)(27 47)(30 75)(31 43)(33 72)(34 38)(36 58)(37 68)(40 62)(41 78)(42 56)(44 48)(45 74)(46 52)(49 53)(50 77)(54 73)(59 63)(66 70)

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)

(1 75)(2 80)(3 77)(4 74)(5 79)(6 76)(7 73)(8 78)(9 24)(10 21)(11 18)(12 23)(13 20)(14 17)(15 22)(16 19)(25 70)(26 67)(27 72)(28 69)(29 66)(30 71)(31 68)(32 65)(33 47)(34 44)(35 41)(36 46)(37 43)(38 48)(39 45)(40 42)(49 59)(50 64)(51 61)(52 58)(53 63)(54 60)(55 57)(56 62)

G:=sub<Sym(80)| (1,79)(2,80)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19)(25,70)(26,71)(27,72)(28,65)(29,66)(30,67)(31,68)(32,69)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,57)(56,58), (1,58,15,40,67)(2,33,59,68,16)(3,69,34,9,60)(4,10,70,61,35)(5,62,11,36,71)(6,37,63,72,12)(7,65,38,13,64)(8,14,66,57,39)(17,29,55,45,78)(18,46,30,79,56)(19,80,47,49,31)(20,50,73,32,48)(21,25,51,41,74)(22,42,26,75,52)(23,76,43,53,27)(24,54,77,28,44), (1,71)(2,12)(3,64)(4,39)(5,67)(6,16)(7,60)(8,35)(9,65)(10,57)(11,15)(13,69)(14,61)(17,51)(18,22)(19,76)(20,28)(21,55)(23,80)(24,32)(25,29)(26,79)(27,47)(30,75)(31,43)(33,72)(34,38)(36,58)(37,68)(40,62)(41,78)(42,56)(44,48)(45,74)(46,52)(49,53)(50,77)(54,73)(59,63)(66,70), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80), (1,75)(2,80)(3,77)(4,74)(5,79)(6,76)(7,73)(8,78)(9,24)(10,21)(11,18)(12,23)(13,20)(14,17)(15,22)(16,19)(25,70)(26,67)(27,72)(28,69)(29,66)(30,71)(31,68)(32,65)(33,47)(34,44)(35,41)(36,46)(37,43)(38,48)(39,45)(40,42)(49,59)(50,64)(51,61)(52,58)(53,63)(54,60)(55,57)(56,62)>;

G:=Group( (1,79)(2,80)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19)(25,70)(26,71)(27,72)(28,65)(29,66)(30,67)(31,68)(32,69)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,57)(56,58), (1,58,15,40,67)(2,33,59,68,16)(3,69,34,9,60)(4,10,70,61,35)(5,62,11,36,71)(6,37,63,72,12)(7,65,38,13,64)(8,14,66,57,39)(17,29,55,45,78)(18,46,30,79,56)(19,80,47,49,31)(20,50,73,32,48)(21,25,51,41,74)(22,42,26,75,52)(23,76,43,53,27)(24,54,77,28,44), (1,71)(2,12)(3,64)(4,39)(5,67)(6,16)(7,60)(8,35)(9,65)(10,57)(11,15)(13,69)(14,61)(17,51)(18,22)(19,76)(20,28)(21,55)(23,80)(24,32)(25,29)(26,79)(27,47)(30,75)(31,43)(33,72)(34,38)(36,58)(37,68)(40,62)(41,78)(42,56)(44,48)(45,74)(46,52)(49,53)(50,77)(54,73)(59,63)(66,70), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80), (1,75)(2,80)(3,77)(4,74)(5,79)(6,76)(7,73)(8,78)(9,24)(10,21)(11,18)(12,23)(13,20)(14,17)(15,22)(16,19)(25,70)(26,67)(27,72)(28,69)(29,66)(30,71)(31,68)(32,65)(33,47)(34,44)(35,41)(36,46)(37,43)(38,48)(39,45)(40,42)(49,59)(50,64)(51,61)(52,58)(53,63)(54,60)(55,57)(56,62) );

G=PermutationGroup([[(1,79),(2,80),(3,73),(4,74),(5,75),(6,76),(7,77),(8,78),(9,20),(10,21),(11,22),(12,23),(13,24),(14,17),(15,18),(16,19),(25,70),(26,71),(27,72),(28,65),(29,66),(30,67),(31,68),(32,69),(33,47),(34,48),(35,41),(36,42),(37,43),(38,44),(39,45),(40,46),(49,59),(50,60),(51,61),(52,62),(53,63),(54,64),(55,57),(56,58)], [(1,58,15,40,67),(2,33,59,68,16),(3,69,34,9,60),(4,10,70,61,35),(5,62,11,36,71),(6,37,63,72,12),(7,65,38,13,64),(8,14,66,57,39),(17,29,55,45,78),(18,46,30,79,56),(19,80,47,49,31),(20,50,73,32,48),(21,25,51,41,74),(22,42,26,75,52),(23,76,43,53,27),(24,54,77,28,44)], [(1,71),(2,12),(3,64),(4,39),(5,67),(6,16),(7,60),(8,35),(9,65),(10,57),(11,15),(13,69),(14,61),(17,51),(18,22),(19,76),(20,28),(21,55),(23,80),(24,32),(25,29),(26,79),(27,47),(30,75),(31,43),(33,72),(34,38),(36,58),(37,68),(40,62),(41,78),(42,56),(44,48),(45,74),(46,52),(49,53),(50,77),(54,73),(59,63),(66,70)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80)], [(1,75),(2,80),(3,77),(4,74),(5,79),(6,76),(7,73),(8,78),(9,24),(10,21),(11,18),(12,23),(13,20),(14,17),(15,22),(16,19),(25,70),(26,67),(27,72),(28,69),(29,66),(30,71),(31,68),(32,65),(33,47),(34,44),(35,41),(36,46),(37,43),(38,48),(39,45),(40,42),(49,59),(50,64),(51,61),(52,58),(53,63),(54,60),(55,57),(56,62)]])

56 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	2J	2K	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	5	8A	···	8P	10A	···	10G	20A	···	20H
order	1	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4	4	4	4	4	5	8	···	8	10	···	10	20	···	20
size	1	1	1	1	2	2	5	5	5	5	10	10	1	1	1	1	2	2	5	5	5	5	10	10	4	10	···	10	4	···	4	4	···	4

56 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	4	4	4	4
type	+	+	+	+	+	+					+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	M₄(2)	F₅	C₂×F₅	C₂×F₅	D₅⋊M₄(2)
kernel	C₂×D₅⋊M₄(2)	C₂×D₅⋊C₈	C₂×C₄.F₅	D₅⋊M₄(2)	C₂×C₂².F₅	D₅×C₂²×C₄	C₂×C₄×D₅	C₂²×C₂₀	C₂³×D₅	D₁₀	C₂²×C₄	C₂×C₄	C₂³	C₂
# reps	1	2	2	8	2	1	12	2	2	8	1	6	1	8

Matrix representation of C₂×D₅⋊M₄(2) ►in GL₆(𝔽₄₁)

40	0	0	0	0	0
0	40	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	40	0	0
0	0	1	6	0	0
0	0	0	0	40	35
0	0	0	0	6	35

1	0	0	0	0	0
0	1	0	0	0	0
0	0	35	6	0	0
0	0	1	6	0	0
0	0	0	0	6	1
0	0	0	0	6	35

40	2	0	0	0	0
4	1	0	0	0	0
0	0	0	0	40	0
0	0	0	0	0	40
0	0	32	28	0	0
0	0	0	9	0	0

40	0	0	0	0	0
40	1	0	0	0	0
0	0	40	0	0	0
0	0	0	40	0	0
0	0	0	0	1	0
0	0	0	0	0	1

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,6,0,0,0,0,0,0,40,6,0,0,0,0,35,35],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,35,1,0,0,0,0,6,6,0,0,0,0,0,0,6,6,0,0,0,0,1,35],[40,4,0,0,0,0,2,1,0,0,0,0,0,0,0,0,32,0,0,0,0,0,28,9,0,0,40,0,0,0,0,0,0,40,0,0],[40,40,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₂×D₅⋊M₄(2) in GAP, Magma, Sage, TeX

C_2\times D_5\rtimes M_4(2)

% in TeX

G:=Group("C2xD5:M4(2)");

// GroupNames label

G:=SmallGroup(320,1589);

// by ID

G=gap.SmallGroup(320,1589);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,184,1123,102,6278,818]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^5=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,d*b*d^-1=b^3,b*e=e*b,d*c*d^-1=b^2*c,c*e=e*c,e*d*e=d^5>;

// generators/relations

G = C2×D5⋊M4(2) order 320 = 26·5

Direct product of C2 and D5⋊M4(2)

G = C₂×D₅⋊M₄(2) order 320 = 2⁶·5

Direct product of C₂ and D₅⋊M₄(2)